Properly twisted groups and their algebras

Abstract

A proper twist on a group G is a function α : G × G → {−1, 1} with the property that, if p, q ∈ G then α(p, q) α(q, q −1 ) = α(pq, q −1 ) and α(p −1 , p) α(p, q) = α(p −1 , pq). The span V of a set of unit vectors B = {ip | p ∈ G} over a ring K with product xy = p,q α(p, q)xpyqipq is a twisted group algebra. If the twist α is proper, then the conjugate defined by x * = p α(p, p −1 )x * p i p −1 and inner product x, y = p xpy * p , satisfy the adjoint properties xy, z = y, x * z and x, yz = xz * , y for all x, y, z ∈ V . Proper twists on Z N 2 over the reals produce the complex numbers, quaternions, octonions and all higher order Cayley-Dickson and Clifford algebras.

A catalog of Cayley-Dickson-like products

“…For each of the eight Cayley-Dickson products, the sense of the three sides is the sameeither clockwise (←) around the triangle, or counter-clockwise (→). If clockwise, then the three sides of the triangle represent (5, 2, 7), (7,4,3) and (3,6,5). If counter-clockwise then the three sides represent (7, 2, 5), (5,6,3) and (3,4,7).…”

“…If clockwise, then the three sides of the triangle represent (5, 2, 7), (7,4,3) and (3,6,5). If counter-clockwise then the three sides represent (7, 2, 5), (5,6,3) and (3,4,7). The circle through the midpoints of the sides represents either (2,4,6) in the clockwise sense ( ) or (2,6,4) in the counter-clockwise sense ( ).…”